Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces

We give an embedding of the Poincaré halfspace $H^D$ into a discrete metric space based on a binary tiling of $H^D$, with additive distortion $O(\log D)$. It yields the following results. We show that any subset P of n points in $H^D$ can be embedded into a graph-metric with $2^{O\left(D\right)}n$ vertices and edges, and with additive distortion $O(\log D)$. We also show how to construct, for any k, an $O(k\log D)$-purely additive spanner of P with $2^{O\left(D\right)}n$ Steiner vertices and $2^{O\left(D\right)}n\cdot {\lambda }_k\left(n\right)$ edges, where ${\lambda }_k\left(n\right)$ is the kth-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for P of size $2^{O\left(D\right)}n$. It allows us to answer approximate near-neighbor queries in $2^{O\left(D\right)}+O(D\log n)$ time, with additive error $O(\log D)$. These constructions can be done in $2^{O\left(D\right)}n\log n$ time.